The last two decades have seen major changes in the application of the finite element method. During the seventies the range of elements and their performance was improved. Application of the method was still fairly new and the learning curve was steep. Much of the change, particularly in the functionality of programs, was influenced by response from the growing number of users. The advances in the eighties were greatly influenced by the increased power and reduced cost of computers, and the development of computer graphics.
Development of solution techniques slowed, but powerful graphics-based techniques for the modeling of complex structures were developed. Computer power hungry techniques for the solution of nonlinear problems became a practical reality. Development in the nineties has been dominated by the automation of the modeling process, involving adaptive mesh refinement and design optimization. These have led to a return to the theory to establish error estimating techniques.
The Finite Element (FE) method has been used with great success to solve many types of problems including:
stress analysis PA1 dynamic response PA1 heat transfer PA1 line elements, such as beams, struts and pipes PA1 two-dimensional solid elements for modeling plane strain and axisymmetry PA1 three-dimensional solid elements for modeling any solid shape PA1 shell elements suitable for modeling thin structures. PA1 optimization of product performance PA1 optimization of product cost PA1 reduction of development time PA1 elimination of reduction of testing PA1 achievement of required quality first time PA1 improved safety PA1 satisfaction of design codes PA1 improved information for engineering decisions PA1 fuller understanding of components allowing more rational design
In recent years there has been an increasing requirement for structures and safety related equipment to be seismically qualified. Smaller items can be tested on shake tables on which they are subjected to dynamic loading, simulating the effect of an earthquake. Large structures cannot undergo this type of test and it is necessary to simulate the behavior analytically, often using the FE technique.
Destructive testing and service failures often provide very limited information. They may show the weakest part of a structure but give no clear indication of the most effective remedial action, where another failure might occur after redesign, or where material could be removed without detriment.
A similar approach was adopted in the analysis of certain types of structures, such as building frames, process piping and aircraft structures. The structure was broken down into smaller parts or elements for which exact results were available, and the behavior of the structure as a whole found by the solution of a set of simultaneous equations.
Although originally intended to represent sections of structures which were in reality quite discrete, the early elements were used with considerable success to analyze more general structures. Modern, general-purpose finite element programs have libraries of elements which permit many different geometries to be modeled. They can be broken down into four main categories:
Common finite elements are illustrated in FIGS. 1A-1D, and examples of their use in structures are shown in FIGS. 2A-2D.
Three-dimensional solid elements are probably the simplest to understand. As in the earlier methods described above, the structure is represented by an assemblage of notional elements, which, for the sake of visualization, may be likened to building blocks.
Unlike an element that is exact, the stiffness of solid elements are calculated approximately by numerical integration, based on assumptions about how the element deforms under loads at the node points. Providing that the elements are sufficiently small, the error due to these approximations is acceptable.
For design by analysis it is necessary to postulate sizes and thicknesses first, and then calculate the stresses. The process is frequently iterative, with changes of size or thicknesses in each loop until a satisfactory or optimized design is achieved. The finite element method lends itself readily to this process since it is often possible to change just a single or few numerical values in the input data to change sizes or thicknesses.
One computer simulation process used for structural design is the use of tetrahedral elements in the finite element analysis and simulation of structures. This is particularly desirable because automatic mesh-generation techniques are now available to subdivide general objects of any shape into meshes of tetrahedral elements. Second-order tetrahedral elements usually give accurate results in small and finite deformation problems with no contact. However, I have discovered that these elements are not appropriate for contact problems because in uniform pressure situations the contact forces are non-uniform at the corner and midside nodes.
I have also discovered that, for second-order tetrahedra, the contact forces at the corner nodes are zero, while the midside nodes carry all the contact load. The zero contact forces at the corner nodes in the tetrahedra result in zero contact pressures. As an additional complication, I have determined that non-convergence of contact conditions may result with second-order tetrahedral elements. I have also discovered that the second-order tetrahedra elements may exhibit significant volumetric locking when incompressibility is approached.
In comparison, the first-order tetrahedra produce uniform contact forces and pressures, but overall results can be very inaccurate due to severe volumetric and shear locking. In addition, very fine meshes may be needed for first-order tetrahedra elements to attain results of sufficient accuracy.
I have also realized that it is desirable to simulate and analyze the structural integrity of structures via computer implemented simulation by prescribing modified second-order tetrahedral elements during small and finite deformation.
I have also realized that it is desirable to simulate and analyze the structural integrity of structures by prescribing modified second-order tetrahedral elements that provide robustness and convergence for contact simulations.
I have also realized that it is desirable to utilize second-order tetrahedra elements that minimize volumetric locking when incompressibility is approached.